A transport plane takes off from a level landing field with two gliders in tow, one behind the other. The mass of each glider is 700 {kg}, and the total resistance (air drag plus friction with the runway) on each may be assumed constant and equal to 2100N . The tension in the towrope between the transport plane and the first glider is not to exceed 12000 ( N}.
A)If a speed of 40 { m/s} is required for takeoff, what minimum length of runway is needed?
B)What is the tension in the towrope between the two gliders while they are accelerating for the takeoff?
A)
2ma=F-2T = >
a=(F-2T)/2m = (12000-2•2100)/2•700 = 5.57 m/s²
s=v²/2a = 40²/2•5.57 =143.6 m.
B)
T₁=ma +F(fr) =700•5.57 + 2100 =5999 N
Elena is correct. No reason for the thumbs down. Just plug in your own values as they're most likely different.
Don't forget to put a parentheses between the / and the 2 and one at the end if you using a calculator. I think that's messing people up.
A) Well, this is some serious towing business we got here. So, let's start with some math magic. We know that the total resistance on each glider is 2100N, and we want to find the minimum length of the runway needed. To figure this out, we need to consider the force required to overcome the resistance. Since we're dealing with gliders, we don't have any engine power to help us out.
Now, the force required to overcome the resistance is given by the equation: Force = Mass x Acceleration. In this case, the acceleration is the change in velocity over time, which we can think of as the takeoff speed divided by the time taken for takeoff.
Since we want to find the minimum length of the runway, we can assume that the acceleration is constant. Using the equation v = u + at (where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken), we can rearrange it to find the time taken for takeoff: t = (v - u) / a.
Now, the initial velocity is 0 m/s (since we're starting from rest), and the final velocity is 40 m/s. The acceleration can be found using the force equation: Force = Mass x Acceleration. Rearranging it, we get: Acceleration = Force / Mass. Plugging in the numbers, we get Acceleration = 2100 N / 700 kg.
Now, we can use all these numbers to find the time taken for takeoff. Once we have that, we can find the minimum length of the runway using the equation: Distance = (Initial velocity x Time taken) + (1/2 x Acceleration x Time taken squared). Plugging in the numbers, we'll have our answer.
B) Now, let's talk about the tension in the towrope between the two gliders while they are accelerating for takeoff. This is like a tug of war between the gliders, you know, a bit like sibling rivalry. But instead of pulling each other's hair, they're pulling a towrope.
To find the tension in the towrope, we can use the equation: Tension = Mass x Acceleration. The mass in this case is the sum of the masses of both gliders, which is 1400 kg. The acceleration is the same as we calculated before, which is 2100 N / 700 kg.
So, plug in those numbers into the equation, and you'll have the tension in the towrope while the gliders are accelerating for takeoff. It's all about teamwork, you know. They're in it together, just like good ol' buddies.
To solve this problem, we can use the principles of Newton's laws of motion. Let's break it down into two parts:
A) Minimum length of runway needed for takeoff:
To find the minimum length of runway needed, we need to consider the forces acting on the system and use the equation of motion.
Given:
Mass of each glider (m): 700 kg
Resistance on each glider (R): 2100 N
Required speed for takeoff (v): 40 m/s
Maximum tension in the towrope between the transport plane and the first glider (T_max): 12000 N
1. Identify the forces acting on the system:
a) The force of tension from the towrope between the transport plane and the first glider (T1).
b) The force of tension from the towrope between the first and second gliders (T2).
c) The resistance force acting on each glider (R).
2. Apply Newton's second law of motion to each glider:
For the first glider: ΣF1 = ma1
T1 - R = m1a1
For the second glider: ΣF2 = ma2
T2 - R = m2a2
3. For takeoff, the acceleration of the system is the same:
a1 = a2 = a
4. Rewrite the equations by substituting the given values from step 2 and the acceleration from step 3:
T1 - R = m1a
T2 - R = m2a
a = v / t (where t is the time required for takeoff)
5. Find the tension T1 using the maximum allowable tension value:
T1 ≤ T_max
T1 ≤ 12000 N
6. Solve the equations simultaneously to determine the acceleration (a).
7. Use the acceleration (a) and the required takeoff speed (v) to find the time (t) using the formula: a = v / t.
8. Finally, calculate the minimum length of the runway needed using the formula: distance = v * t.
B) Tension in the towrope between the two gliders while accelerating for takeoff:
To find the tension in the towrope between the two gliders, we need to consider the forces acting on the system and use the equation of motion.
1. Identify the forces acting on the system:
a) The tension force from the transport plane acting on the first glider (T1).
b) The tension force between the first and second gliders (T2).
2. Apply Newton's second law of motion to each glider:
For the first glider: ΣF1 = ma1
T1 - T2 - R = m1a1
For the second glider: ΣF2 = ma2
T2 - R = m2a2
3. Substitute the given values for the mass (m), resistance (R), and acceleration (a) if given.
4. Solve the equations simultaneously to find the tension T2.